## Introduction1

The Moran eigenvector approach involved the spatial patterns represented by maps of eigenvectors; by choosing suitable orthogonal patterns and adding them to a linear or generalised linear model, the spatial dependence present in the residuals can be moved into the model.

It uses brute force to search the set of eigenvectors of the matrix $$\mathbf{M W M}$$, where

$\mathbf{M} = \mathbf{I} - \mathbf{X}(\mathbf{X}^{\rm T} \mathbf{X})^{-1}\mathbf{X}^{\rm T}$ is a symmetric and idempotent projection matrix and $$\mathbf{W}$$ are the spatial weights. In the spatial lag form of SpatialFiltering and in the GLM ME form below, $$\mathbf{X}$$ is an $$n$$-vector of ones, that is the intercept only.

In its general form, SpatialFiltering chooses the subset of the $$n$$ eigenvectors that reduce the residual spatial autocorrelation in the error of the model with covariates. The lag form adds the covariates in assessment of which eigenvectors to choose, but does not use them in constructing the eigenvectors. SpatialFiltering was implemented and contributed by Yongwan Chun and Michael Tiefelsdorf, and is presented in ; ME is based on Matlab code by Pedro Peres-Neto and is discussed in and .

library(spdep)
require("sf", quietly=TRUE)
NY8 <- st_read(system.file("shapes/NY8_utm18.shp", package="spData"))
## Reading layer NY8_utm18' from data source
##   /home/rsb/lib/r_libs/spData/shapes/NY8_utm18.shp' using driver ESRI Shapefile'
## Simple feature collection with 281 features and 17 fields
## Geometry type: POLYGON
## Dimension:     XY
## Bounding box:  xmin: 358241.9 ymin: 4649755 xmax: 480393.1 ymax: 4808545
## Projected CRS: WGS 84 / UTM zone 18N
NY_nb <- read.gal(system.file("weights/NY_nb.gal", package="spData"), override.id=TRUE)
library(spatialreg)
nySFE <- SpatialFiltering(Z~PEXPOSURE+PCTAGE65P+PCTOWNHOME, data=NY8, nb=NY_nb, style="W", verbose=FALSE)
nylmSFE <- lm(Z~PEXPOSURE+PCTAGE65P+PCTOWNHOME+fitted(nySFE), data=NY8)
summary(nylmSFE)
##
## Call:
## lm(formula = Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + fitted(nySFE),
##     data = NY8)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -1.5184 -0.3523 -0.0105  0.3221  3.1964
##
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)
## (Intercept)        -0.51728    0.14606  -3.542 0.000469 ***
## PEXPOSURE           0.04884    0.03230   1.512 0.131717
## PCTAGE65P           3.95089    0.55776   7.083 1.25e-11 ***
## PCTOWNHOME         -0.56004    0.15688  -3.570 0.000423 ***
## fitted(nySFE)vec13 -2.09397    0.60534  -3.459 0.000630 ***
## fitted(nySFE)vec44 -2.24003    0.60534  -3.700 0.000261 ***
## fitted(nySFE)vec6   1.02979    0.60534   1.701 0.090072 .
## fitted(nySFE)vec38  1.29282    0.60534   2.136 0.033613 *
## fitted(nySFE)vec20  1.10064    0.60534   1.818 0.070150 .
## fitted(nySFE)vec14 -1.05105    0.60534  -1.736 0.083662 .
## fitted(nySFE)vec75  1.90600    0.60534   3.149 0.001826 **
## fitted(nySFE)vec21 -1.06331    0.60534  -1.757 0.080138 .
## fitted(nySFE)vec36 -1.17861    0.60534  -1.947 0.052578 .
## fitted(nySFE)vec61 -1.08582    0.60534  -1.794 0.073986 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6053 on 267 degrees of freedom
## Multiple R-squared:  0.3401, Adjusted R-squared:  0.308
## F-statistic: 10.58 on 13 and 267 DF,  p-value: < 2.2e-16
nylm <- lm(Z~PEXPOSURE+PCTAGE65P+PCTOWNHOME, data=NY8)
anova(nylm, nylmSFE)
## Analysis of Variance Table
##
## Model 1: Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME
## Model 2: Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + fitted(nySFE)
##   Res.Df     RSS Df Sum of Sq      F    Pr(>F)
## 1    277 119.619
## 2    267  97.837 10    21.782 5.9444 3.988e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Since the SpatialFiltering approach does not allow weights to be used, we see that the residual autocorrelation of the original linear model is absorbed, or ‘whitened’ by the inclusion of selected eigenvectors in the model, but that the covariate coefficients change little. The addition of these eigenvectors – each representing an independent spatial pattern – relieves the residual autocorrelation, but otherwise makes few changes in the substantive coefficient values.

The ME function also searches for eigenvectors from the spatial lag variant of the underlying model, but in a GLM framework. The criterion is a permutation bootstrap test on Moran’s $$I$$ for regression residuals, and in this case, because of the very limited remaining spatial autocorrelation, is set at $$\alpha = 0.5$$. Even with this very generous stopping rule, only few eigenvectors are chosen; their combined contribution only just improves the fit of the GLM model.

NYlistwW <- nb2listw(NY_nb, style = "W")
set.seed(111)
nyME <- ME(Cases~PEXPOSURE+PCTAGE65P+PCTOWNHOME, data=NY8, offset=log(POP8), family="poisson", listw=NYlistwW, alpha=0.44)
nyME
##   Eigenvector ZI pr(ZI)
## 0          NA NA   0.31
## 1          24 NA   0.44
## 2         223 NA   0.42
## 3         206 NA   0.43
## 4         169 NA   0.48
NY8$eigen_24 <- fitted(nyME)[,1] NY8$eigen_223 <- fitted(nyME)[,2]
nyglmME <- glm(Cases~PEXPOSURE+PCTAGE65P+PCTOWNHOME+offset(log(POP8))+fitted(nyME), data=NY8, family="poisson")
summary(nyglmME)
##
## Call:
## glm(formula = Cases ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + offset(log(POP8)) +
##     fitted(nyME), family = "poisson", data = NY8)
##
## Deviance Residuals:
##     Min       1Q   Median       3Q      Max
## -2.9695  -0.9875  -0.2365   0.6506   3.3866
##
## Coefficients:
##                    Estimate Std. Error z value Pr(>|z|)
## (Intercept)        -8.13431    0.18388 -44.237  < 2e-16 ***
## PEXPOSURE           0.14136    0.03134   4.511 6.45e-06 ***
## PCTAGE65P           4.16875    0.60149   6.931 4.19e-12 ***
## PCTOWNHOME         -0.39290    0.19222  -2.044  0.04096 *
## fitted(nyME)vec24   1.62658    0.72243   2.252  0.02435 *
## fitted(nyME)vec223  0.92941    0.70391   1.320  0.18671
## fitted(nyME)vec206 -0.11559    0.68987  -0.168  0.86693
## fitted(nyME)vec169 -1.82674    0.68142  -2.681  0.00735 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
##     Null deviance: 428.25  on 280  degrees of freedom
## Residual deviance: 340.08  on 273  degrees of freedom
## AIC: Inf
##
## Number of Fisher Scoring iterations: 5
nyGLMp <- glm(Cases~PEXPOSURE+PCTAGE65P+PCTOWNHOME+offset(log(POP8)), data=NY8,family="poisson")
anova(nyGLMp, nyglmME, test="Chisq")
## Analysis of Deviance Table
##
## Model 1: Cases ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + offset(log(POP8))
## Model 2: Cases ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + offset(log(POP8)) +
##     fitted(nyME)
##   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1       277     353.35
## 2       273     340.08  4   13.269  0.01003 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1`

Figure $fig:geigen2$ shows the spatial patterns chosen to match the very small amount of spatial autocorrelation remaining in the model. As with the other Poisson regressions, the closeness to TCE sites is highly significant. Since, however, many TCE sites are also in or close to more densely populated urban areas with the possible presence of both point-source and non-point-source pollution, it would be premature to take such results simply at their face value. There is, however, a potentially useful contrast between the cities of Binghamton in the south of the study area with several sites in its vicinity, and Syracuse in the north without TCE sites in this data set.

## References

Dray, S., P. Legendre, and P. R. Peres-Neto. 2006. “Spatial Modeling: A Comprehensive Framework for Principle Coordinate Analysis of Neighbor Matrices (PCNM).” Ecological Modelling 196: 483–93.
Griffith, D. A., and P. R. Peres-Neto. 2006. “Spatial Modeling in Ecology: The Flexibility of Eigenfunction Spatial Analyses.” Ecology 87: 2603–13.
Tiefelsdorf, M., and D. A. Griffith. 2007. “Semiparametric Filtering of Spatial Autocorrelation: The Eigenvector Approach.” Environment and Planning A 39: 1193–1221.

1. This vignette formed pp. 302–305 of the first edition of Bivand, R. S., Pebesma, E. and Gómez-Rubio V. (2008) Applied Spatial Data Analysis with R, Springer-Verlag, New York. It was retired from the second edition (2013) to accommodate material on other topics, and is made available in this form with the understanding of the publishers.↩︎